Geometric Progression: Sum of n Terms (Finite)

$$a_{1},\dfrac{a_{2}}{2},\dfrac{a_{3}}{2^{2}},....,\dfrac{a_{10}}{2^{9}}$$ be a G.P. of common ratio $$\dfrac{1}{\sqrt{2}}$$

$$\dfrac{\frac{a_2}{2}}{a_1}=\dfrac{1}{\sqrt{2}}$$ => $$a_2=\sqrt{2}a_1$$

$$\dfrac{\frac{a_3}{2^2}}{\frac{a_2}{2}}=\dfrac{1}{\sqrt{2}}$$ => $$a_3=\sqrt{2}a_2$$

Thus, $$a_1,a_2,a_3,....$$ is a GP with a common ratio of $$\sqrt{2}$$

We are given that $$a_{1}+a_{2}+....+a_{10}=62$$.

Sum of GP = $$a\left(\dfrac{r^{10}-1}{r-1}\right)$$

=> $$a_1\left(\dfrac{\left(\sqrt{2}\right)^{10}-1}{\sqrt{2}-1}\right)=62$$

=> $$a_1\left(\dfrac{31}{\sqrt{2}-1}\right)=62$$

=> $$a_1=2(\sqrt{2}-1)$$